Question on linear transformaiton

**noricka** · December 2nd 2011, 01:45 AM

Let V and W be vector spaces over F field, dimV=n, and dim W=m. What is the dimension of the vector space L(V,W), and what is an optional basis?

Thank you!

**FernandoRevilla** · December 2nd 2011, 02:05 AM

Originally Posted by noricka

Let V and W be vector spaces over F field, dimV=n, and dim W=m. What is the dimension of the vector space L(V,W), and what is an optional basis?

Hint : If we fix $B_V$ and $B_W$ basis of $V$ and $W$ respectively, there is a natural isomorphism $\phi:\mathcal{L}(V,W)\to \mathbb{K}^{m \times n}$ given by $\phi (T)=[T]_{B_V}^{B_W}$ .

**Deveno** · December 2nd 2011, 11:28 AM

you may also wish to consider $E_{i,j} \in \mathcal{L}(V,W) : E_{i,j}(a_1v_1+a_2v_2+\dots+a_nv_n) = a_jw_i$ .

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